Newton's Three Laws of Motion

Four thousands of years since Aristotle's time, we thought that things that are stationary tend to stay stationary and that things that are moving tend to slow down. This is reasonable since it agrees with out everyday experiences. For example if I push a rock or throw something and it starts moving, it will tend to slow down and eventually come to rest. Isaac Newton stated his first law of motion in his Principia. This law states that if an object is moving at a constant velocity $$\vec{v}$$, it will tend to keep moving in a straight line at the constant velocity $$\vec{v}$$ unless it is acted upon by an external force. Contrary to what Aristotle and his successors believed, objects moving tend to keep on moving!

For example, if I threw a rock in a cosmic void (vast regions of empty space which can extend for millions of light years), that rock would move in a straight line at a constant velocity $$\vec{v}$$ practically forever—at least until it reached the nearest galaxy. Also, if you instead  simply just let go of the rock without budging it, its velocity would stay as $$\vec{v}=0$$ practically forever.

But why is it that if I threw a rock on Earth's surface, it would eventually slow down and hit the ground? To answer this question, we need to use Newton's second law of motion which he published  in his Principia which states

$$\sum{\vec{F}}=m\vec{a},$$

where $$\sum{\vec{F}}$$ is the total force acting on the object, $$m$$ is the object's mass, and $$\vec{a}$$ is the object's acelleration. All objects which we'll be encountering in the next several lessons have some positive amount of mass. Thus, according to the equation $$\sum{\vec{F}}=m\vec{a}$$ where $$\vec{a}=\frac{d\vec{v}}{dt}$$, the only way that an object can acellerate is if some net force is acting upon it. And when an object acellerates by an amount $$\vec{a}$$, that object's velocity $$\vec{v}$$ will change with time according to the derivitive $$\frac{d}{dt}\vec{v}$$. The reason why when object's are thrown on Earth they slow down and eventually come to rest is because forces are acting upon them. These forces include the Earth's gravity and also the force due to air friction.

Newton's second law, at first glance, might appear to lead to a contradiction. When you push on a brick wall, you are clearly applying a force to that wall. Why doesn't that wall accellerate then? This question can be answer using Newton's third law which states that:

Whenever an object of mass $$m_1$$ applies a force $$\vec{F}_{1,2}$$ to a mass $$m_2$$, the object of mass $$m_2$$ applies an equal-and-opposite force $$\vec{F}_{2,1}=-\vec{F}_{1,2}$$ on the object of mass $$m_1$$.

Therefore, according to Newton's third law, if I push on a wall with a force of $$+5N$$ to the right, the wall will exert an equal-and-opposite force of $$-5N$$ on my hand to the left. Since the net force on the hand-wall system is zero, the net acceleration $$\vec{a}$$ of the system is zero. Thus, both your hand and the wall remain stationary.